Newton, Dios y el caos

Alejandro Jenkins, UCR

III Mini-congreso, IFT19 dic. 2013

Isaac Newton (1642-1727)

• Lagrange: el mayor y más afortunado de los mortales, porque solo se puede descubrir una vez el sistema del mundo

• Weinberg: “al educar a los nuevos físicos lo primero que les enseñamos sigue siendo la mecánica newtoniana, y nunca olvidan cómo pensar en términos newtonianos”

Busto por L.-F. Roubillac. Fuente: Foster (1928)

Newton, cont.

• Newton: “No sé cómo me vea el mundo, pero yo me veo como un niño jugando en playa, que se divierte al encontrar de vez en cuando una piedra más lisa o una concha más bonita de lo usual, mientras que el gran océano de la verdad se extiende desconocido ante mí.”

• Keynes: “el último de los magos”

Estatua por L.-F. Roubillac, capilla de Trinity College, Cambridge. Fuente: Wikipedia

Genealogía

• Newton venía de un trasfondo puritano

• Cromwell, Milton, los Pilgrims

• Cambridge, Harvard, Yale, Princeton

• Weber, Troeltsch, Merton et al. subrayan relación entre puritanismo y revoluciones científica e industrial

Woolsthorpe Manor. Fuente: Flickr

Religión• Ultra-protestante

• Involucrado en “Revolución Gloriosa” de 1688

• Secretamente anti-trinitario (arriano)

• Descreía de inmortalidad del alma

• Dispuesto a perder cátedra antes que ordenarse sacerdote anglicano

• Rechazó extremaunciónhttp://blog.londonconnection.com/wp-content/uploads/2013/05/Picture-92.png

Disputa con Leibniz• Newton descubrió cálculo antes

de 1670, pero lo circuló solo en cartas privadas

• Leibniz publicó resultados equivalentes a partir de 1684

• Relación con Newton fue buena hasta ca. 1700

• A.R. Hall: “Leibniz no era una persona modesta. En esto era realmente distinto a Newton.”

Retrato por C.B. Francke. Fuente: Wikipedia

Disputa con Leibniz, cont.

• Pleito por historia del cálculo alcanza apogeo en 1710-12

• Leibniz rechaza gravedad newtoniana

• alega que constituye “causa oculta”

• niega que tiempo y espacio puedan definirse sin referencia a la materia (Berkeley, Mach)

• niega posibilidad del vacío y existencia de átomos

• intenta formulación de mecánica basada en conservación de vis viva (energía)

Principia, 2da ed.• Aquel tan presuntuoso como para creer que puede

hallar los verdaderos principios de la física y las leyes naturales por la mera fuerza de su propia mente y por la luz interna de la razón, debe suponer, o bien que el mundo existe por necesidad y que por la misma necesidad sigue las leyes propuestas, o, si el orden la naturaleza fue establecido por la voluntad de Dios, que él mismo, un miserable reptil, puede decir qué es lo más adecuado.

• -Introducción de Roger Cotes (1713)

Leibniz a Carolina (nov. 1715)• Newton y sus seguidores tienen una

opinión muy curiosa del trabajo de Dios. Según ellos, el reloj de Dios, el Universo, dejaría de funcionar si no le diera cuerda de vez en cuando. No tuvo suficiente previsión para hacerlo un movimiento perpetuo. La máquina que ha construido es tan imperfecta que necesita repararla de tiempo en tiempo por una intervención milagrosa, e inclusive repararla, como un relojero repara su aparato.

Retrato de J. Amigoni (1735). Fuente: Wikipedia

Opticks, “Query 31” (1706)

• Mientras que los cometas se mueven en órbitas muy excéntricas y en toda clase de posiciones, el mero azar no hubiera hecho que los planetas se movieran todos en órbitas concéntricas, exceptuando algunas pequeñas irregularidades, que pueden haber sido producidas por las acciones mutuas de los cometas y planetas y que bien podrían tender a aumentar, hasta que el sistema requiera una reforma.

Precesión lunar• Conocida desde la

antigüedad

• plano orbital inclinado ∼ 5º respecto a eclíptica

• nodos precesan hacia atrás con período ∼ 18 años

• ábsides precesan hacia adelante con período ∼ 8 años

http://www.chara.gsu.edu/lab/labs/lab4.html

Fuente: Bondemann (2010)

Meses

• Sinódico: 29,53059 d

• Sideral: 27,32166 d

• Tropical: 27,32158 d

• Anomalístico: 27,55455 d

• Draconítico: 27,21222 d

Fuente: Wikipedia

Desigualdades lunares

• anomalía (babilonios): A ∼ 6º 15’ , T = 1 mes

• evección (Ptolomeo): A ∼ 1º 15’ , T ∼ 32 días

• variación (Tycho): A ∼ 40’ , T = 1/2 mes

• ec. anual (Tycho y Kepler): A ∼ 11’ , T = 1 año

• desigualdad paraláctica (Halley y Newton): A ∼ 2’ , T = 1 mes

f5l 12´ sin l 154

´

2 sin 2l 1¯ . (11)

These formulas will eventually reappear when the mo-tion of the Earth around the Sun is taken into accountexplicitly to get the corrections for the motion of theMoon around the Earth.

Finally, we list two more expressions for the same ex-pansions in order to make the connection with the epi-cycle theory,

1a

~

x1iy!

5ra

ejf5eil F1212

´

2232

´e2il 112

´eil

118

´

2e22il 138

´

2e2il 1¯ G . (12)

The simple epicycle idea has become a Fourier expan-sion. Equivalently,

1a

~

x1iy!

5ra

eif5eil F1212

´

22´ cos l 12i´ sin l

112

´

2 cos 2l 1i4

´

2 sin 2l 1¯ G , (13)

where the real part indicates the correction in the dis-tance r , while the imaginary part describes the correc-tion in the true anomaly f . In lowest order, the latter isobviously twice as large as the former. The factor 2, be-tween 2´ cosl and 2i´ sin l , is designed to preservethe angular momentum and was correctly given in Ptole-my’s equant model.

V. THE MANY MOTIONS OF THE MOON

A. The traditional model of the Moon

A plane through the center of the Earth is determinedat an inclination g of about 5 degrees with respect to theecliptic. The Moon moves around the Earth in thatplane on an ellipse with fixed semi-major axis a and ec-centricity ´ of about 1/18. The Greek model was quitesimilar, except that the ellipse was replaced by an eccen-tric circle.

The plane itself rotates once every 18 years in thebackward direction, i.e., against the prevailing motion inthe solar system, while keeping its inclination constant.The perigee of the Moon, its point of closest approach tothe Earth, makes a complete turn in the forward direc-tion in about nine years.

The following picture (see Fig. 1) emerges: first we fixthe direction of the spring equinox or some fixed starnear it as the universal reference Q in the ecliptic:counting always from west to east, we determine theangle h from Q to the ascending node, i.e., the line ofintersection for the Moon’s orbit with the ecliptic wherethe Moon enters the upper side of the ecliptic; fromthere we move by an angle g in the Moon’s orbital planeuntil we meet the perigee of the Moon; and finally weget to the Moon by moving through the true anomaly f .All these three angles have a double time dependence:

linear (increasing for f and g , while decreasing for h)plus various periodic terms that average to 0.

B. The osculating elements

Assuming that the Moon’s position x* and its momen-tum p* with respect to the Earth are known at sometime t , its total energy (kinetic plus potential with re-spect to the Earth) gives the semi-major axis a ; its an-gular momentum L* yields not only the inclination g andorientation h of its orbital plane, but also its eccentricity´; finally, its so-called Runge-Lenz vector,

F*5@

p* , L*#

1G0EM2 x*/r , (14)

gives the location of the perigee, i.e., the angle g withrespect to the node, and from there the true anomaly f .The masses of the Earth and of the Moon are called Eand M , while G0 is the gravitational constant.

These elements a , ´, g, h , g , f for the Moon give theparameters of the Kepler ellipse that fits the lunar tra-jectory most closely at the time t ; they have complicatedvariations with the time t . The linearly increasing partsin the angles get special names and symbols; they areused as the basis for all the future computations. Themean anomaly l is the linearly varying part of the trueanomaly f ; the mean argument of the latitude F is thelinear part of the distance f1g from the node; the meanlongitude l is the linear part of the distance f1g1hfrom the reference Q .

Whereas a single variable, the mean longitude, is suf-ficient for describing the complete motion of a planetaround the Sun, three angles are required for the Moon:the mean longitude l for the main motion around theEarth, the mean argument of latitude F for the motionout of the ecliptic, and the mean anomaly l for the ra-dial motion. The osculating elements a , g, and ´ haveonly periodic variations. But, according to Fig. 2, it isquite misleading to think of the lunar orbit as having afixed eccentricity like ´>1/18, because the value of theosculating eccentricity varies enormously and quite fast.

FIG. 1. The basic lunar model from antiquity (adopted eversince) consists of an orbital plane for the Moon containing anepicycle for its orbit; the crucial parameters a ,´ ,g and thethree angles l ,g ,h have their modern interpretation.

600 Martin C. Gutzwiller: The oldest three-body problem

Rev. Mod. Phys., Vol. 70, No. 2, April 1998

Fuente: Gutzwiller (1998)

C>2pS 113v

2

2n2 D , (19)

provided the second term is small. In the case of theMoon, the angular speed is given by n52p/T1 , whereT1 is the sidereal month.

Without any explanation, Newton now proposes forthe ratio v

2/n2 the value 100/35745 and concludes thatthe perigee advances by 1°318289. Only the third editionof the Principia, almost 40 years later, ends this wholesection with the cryptic remark: ‘‘The apse of the Moonis about twice as swift.’’ Nevertheless, after the data hadbeen known with high accuracy for over 2000 years,Newton was able to reduce the difference between theanomalistic and the sidereal month to the ratio(T1 /T0)2. The missing factor 2 in the motion of theMoon’s perigee was finally explained 20 years after hisdeath.

E. Proposition LXVI and its 22 corollaries

Modern language and contemporary symbols like vec-tors will be used in this section to explain the basic ideasin proposition LXVI. Nevertheless, it is a sacred duty toemphasize what we owe to Newton’s crucial insights,without which there would be no progress in under-standing the motion of three masses that interactthrough gravitational forces. All his reasoning over 15pages is based on a single diagram (Fig. 3), which isrepeated seven times, on every second page, to spare thereader the tedium of turning pages while thinking. The

corresponding Fig. 4 will be used as the modern equiva-lent, where the nomenclature agrees with the later chap-ters.

The main proposition as applied to the lunar problemclaims the following: The motion of the Moon around theEarth under the gravitational attraction of both the Earthand the Sun comes closer to the ideal Kepler motion, ifthe Earth in turn is allowed to move under the gravita-tional attraction of both the Sun and the Moon ratherthan being forced to remain at a fixed position.

In Fig. 3 we have the Earth with mass E in T (tellus),the Sun with mass S in S (sol), and the Moon with massM in P (planeta). In Fig. 4 the Moon’s and Earth’s loca-tions are designated by M and E , while G is the center ofmass for the double planet Earth-Moon. The vectorsconnecting various points in Fig. 4 are xY of length rpointing from E to M ,xY 8 of length r8 pointing from S toE , j of length r pointing from S to M , and XY of lengthR pointing from S to G.

The crux of Newton’s argument lies in representingthe ‘‘accelerative force’’ of the Sun on the Moon by thesegment SL in Fig. 3. This segment is then decomposedinto the sum of the two segments SM and ML, whereML is parallel to the line TP which connects the Earthwith the Moon. Referring to Fig. 4, we get the first com-ponent of the Moon’s perturbation by the Sun,

G0SxY 8S 1r

321

r83D>3G0SxY 8r8

~

xY 8,xY!

r84 , (20)

which is clearly of the same order as the second compo-nent,

2G0SxY

r

3 >2G0SxY

r83 . (21)

Newton now studies the effect of these perturbativeaccelerations on the orbit of the Moon around theEarth. According to Corollary 2, the Moon’s angularmomentum with respect to the Earth is larger in thesyzygies (new and full moon) than in the quadratures(half moons). Moreover, according to Corollary 3, theMoon’s speed is larger in the syzygies than in thequadratures. Corollary 4 points out, however, that thelunar orbit is more highly curved in the quadratures thanin the syzygies because the solar perturbation decreasesthe Earth’s attraction in the syzygies, while its radialcomponent helps in the quadratures. Corollary 5 ex-

FIG. 3. Newton’s Proposition LXVI from the first book of thePrincipia explains the essence of his lunar theory (see the quo-tation in Sec. VI.A); its 22 corollaries cover 15 pages, and con-tain a single figure, which is repeated on every second page inthe third edition. The Sun’s attraction to the Moon is repre-sented by the line SL and is decomposed into the sum SM1ML for the detailed discussion.

FIG. 4. Modern version of Fig. 3 containing the center of massG of the Earth-Moon double planet that is shown to circle theSun almost exactly on a Kepler ellipse. The vectors are xY fromthe Earth in E to the Moon in M , X* from the Sun in S to G,xY 8 from S to E , and j

* from S to M .

605Martin C. Gutzwiller: The oldest three-body problem

Rev. Mod. Phys., Vol. 70, No. 2, April 1998

Tres cuerpos• Newton estudia órbita lunar en

libros I y III de Principia

• Resuelve completamente variación, ec. anual, precesión de nodos, y oscilación de inclinación

• “una de las partes más profundas de este admirable trabajo” -Laplace (1825)

• “el capítulo más valioso que se ha escrito en las ciencias físicas” -Airy (1834)Principia, I (1687)

Discrepancias• Principia no da

tratamiento claro de evección

• Explica solo 1/2 de precesión de ábsides

• Dudas sobre exactitud de F = G M m / r2

• Despejadas por Clairaut en 1749, yendo a siguiente orden en perturbación

Grabado por L.-J. Cathelin. Fuente: Wikipedia

12/2/13 1:35 PMA new and most accurate theory of the moon's motion: whereby all her irregularities may be solved, .…en by ... Mr. Isaac Newton, and published in Latin by Mr. David Gregory in his excellent Astronomy.

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“Teoría lunar”• Utilidad de tablas lunares para

navegación (problema de longitud)

• En 1702, Newton publica algoritmo con precisión de 2’

• “[Es] desafortunado que un planeta tan cercano como la luna tenga una órbita tan irregular que es, en cierto sentido, vano depender de cualquier cálculo, por más exacto que sea.”

• Gutzwiller (1998): “¡maravillosa definición de lo que ahora llamaríamos caos!”

Perturbaciones• Disputa con Flamsteed por acceso a datos del

observatorio de Greenwich

• Newton desarrolló método equivalente a perturbación de parámetros orbitales, luego redescubierto por Euler, Lagrange y Laplace

• Nunca lo incorporó a Principia

• Desconocido hasta 1872

• Ver Nauenberg (2001)

Mecánica analítica• Desarrollada por Euler,

Lagrange y otros

• Aplica a mecánica newtoniana métodos analíticos de Leibniz

• Arnold: “enseña cómo usar el análisis (y cómo enseñarlo) a personas que no lo entienden”

• Paréntesis de Poisson, transformaciones canónicas, etc., surgen de mecánica celeste

P. S. de Laplace (1749-1827)

• Laplace cree mostrar estabilidad del sistema solar en 1784

• Sobre Dios: “no necesité de esa hipótesis”

• “Demonio de Laplace” (1814)

• Pero su serie no necesariamente converge

Retrato póstumo por J.-B. Paulin Guérin. Fuente: Wikipedia

G. W. Hill (1838-1914)

• ec. de Hill (resonancia paramétrica)

• En 1877, primero en resolver satisfatoriamente precesión de ábsides

• Poincaré: “vemos en esta obra el germen de la mayor parte del progreso que se ha hecho en [la mecánica celeste] desde entonces”

• “esta reserva, iba a decir este salvajismo, ha sido una feliz circunstancia para la ciencia”

q + �2(t)q = 0

Fuente: Collected mathematical works of George William Hill (1905)

u5x1iy , v5x2iy , r25x21y21z25uv1z2,

z5exp~

it!

, D5d

idt

5z

ddz

, (77)

the equations of motion now are

D2u12m8Du2ur3 1

3m82

2 ~

u1v !

50, (78)

D2v22m8Dv2vr3 1

3m82

2 ~

u1v !

50, (79)

D2z2zr32m82z50, (80)

where we have introduced the new parameter m85n8/(n2n8)51/(number of synodic months in oneyear).

Since our dynamic system does not have any time de-pendence, there is an integral of motion C which Hillcalls the Jacobian integral,

12 ~

2DuDv1Dz2!

21r

238

m82~

u1v !

25C . (81)

Normally, one would first of all define the energy of atrajectory by fixing the value of C , but Hill decides tofind a solution that is defined by its period right from thestart, not by its energy.

His ingenious idea is that the real trajectory of theMoon must be close to a periodic orbit with the correctperiod. Among all the known correction terms in theaccepted description of the Moon’s motion, only TychoBrahe’s variation is retained because it has the period ofthe synodic month. The resulting ‘‘variational orbit’’ liesin the ecliptic and has a slightly oval shape which is cen-tered on the Earth, with the long axis in the direction ofthe half moons, as Newton had foreseen.

The required periodicity with the synodic month isenforced by postulating a solution in the form

u0~

t

!

51a

~

x01iy0!

5eitS a01(

j51

`

~

ajzj1a2jz

2j!D ,

(82)

a power series with positive and negative integer expo-nents. If the orbit is symmetric with respect to the x axis,then u0(2t)5u0* , and all the coefficients in the seriesare real. The symmetry with respect to the y axis isequivalent to the point symmetry at the origin, so thatu0(t1p)52u0(t), and all the odd-numbered coeffi-cients vanish.

Hill’s 1878 paper contains a detailed and complete ac-count of his work in finding the variational orbit in theecliptic, z50. The problem is nonlinear because of theterms with 1/r3 in Eqs. (78) and (79). The coefficients inEq. (82) are expanded in powers of m8 to exponentsthat guarantee 15-figure accuracy! Poincare (1907) madeHill’s procedure more transparent in his Lectures on Ce-lestial Mechanics. With the help of the Jacobian integral,Hill was able to manage his computations in rationalarithmetic, and in such a way that he gained a factor m84

at each step.

At the end of this long paper, Hill obtains the varia-tional orbit for increasing values of m8. The number ofsynodic months per year51/m8 comes down from 12.37through the integers until it reaches the critical value1.78265 where the oval shape acquires a cusp in the halfmoons (see Fig. 5). Hill thought the satellite would notreach the half moon position any longer for supercriticalvalues of m8. But Poincare showed that the sequence ofvariational orbits continues with the satellites now mak-ing a short regressive motion around the half moon inthe rotating frame, before taking up again the principalmotion in the forward direction.

A recursive computation for the variational orbit wasset up by Schmidt (1995) with relative ease, provided theprogram can handle rational arithmetic. The expansionof u0 is made in powers of m8 where each term is a finitepolynomial in z and z

21 whose order does not exceedthe power of m8. Each next-higher order follows fromthe lower-order terms in a standard recursion applied toEq. (78).

D. The motion of the lunar perigee

The real trajectory of the Moon is now constructed asa small displacement from the variational orbit. Tomake the distinction, coordinates of the variational orbit

FIG. 5. Hill’s diagram of the variational orbit for differentlengths of the synodic month: a512.369 lunations per year,like our Moon, b54, c53, d51.78265 lunations leading to acusp. Poincare showed that this series continues smoothly,leading to an ever increasing loop around the half moon.

621Martin C. Gutzwiller: The oldest three-body problem

Rev. Mod. Phys., Vol. 70, No. 2, April 1998

Órbita variacional• Dificultad está en no

integrabilidad del problema de tres cuerpos

• No hay soluciones periódicas exactas

• Necesitamos órbita cerrada para definir ábsides

• Hill la obtiene imponiendo periodicidad observada (mes sinódico)

• Perturbación sobre esta órbita variacional da precesión

a = 12.369 meses / año; b = 4; c= 3; d = 1,78265Fuente: Gutzwiller (1998)

H. Poincaré (1854-1912)• Les méthodes nouvelles de la

mécanique céleste (1899)

• “teoría cualitativa de las ecuaciones diferenciales”

• topología diferencial, sistemas dinámicos

• cibernética, catástrofes, caos, complexología...

• ver Holmes (1990)

Teoría del caos

• Facilidad de simulación numérica, a partir de 1950s

• “atractor de Lorenz” (1963)

• caos acuñado por Jim Yorke (1975)

• gran éxito popular: J. Gleick, Chaos: Making of a New Science (1987)

• M. Crichton, Jurassic Park (1990)Fuente: Wikipedia

Huecos de Kirkwood

Mapa de la distribución de asteroides

Fuente: Wikipedia

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5Semi-major Axis (AU)

0

50

100

150

200

250

300

350

Ast

eroi

ds (p

er 0

.000

5 A

U b

in)

3:1 5:2 7:3 2:1

Alan Chamberlin (2007, JPL/Caltech)

Mean Motion Resonance( asteroid : Jupiter )

Asteroid Main-Belt DistributionKirkwood Gaps

Huecos de Kirkwood, cont.

• Wisdom et al. (1984) muestran que órbitas cercanas a resonancias bajas son caóticas

• Tamaño y eccentricidad varían fuertemente

• En una excursión, asteroide puede chocar con Marte

• Ver Arnold (1990) A. Chamberlin (2007), JPL/Caltech

Anillos de Saturno

Satélite Cassini (NASA)

Estabilidad del sistema solar

• Posiciones de planetas son caóticas

• Mover lápiz sobre mi escritorio corre Júpiter de un lado al otro del sol en ∼ 109 años

• Forma de órbitas planetarias parece estable en simulaciones, para vida esperada del sol (8 × 109 años)

• 1% de posibilidad de que Mercurio choque con Venus

• 0,002% de que la Tierra sea destruida

• ver Tremaine (2011)

Principio antrópico• Planeta adicional en casi

cualquier posición causaría inestabilidad inmediata

• No creemos que indique diseño divino, porque hay muchos sistemas solares

• Aplicación de principio antrópico (B. Carter)

B. Fontenelle, Entretiens sur la pluralité des mondes (1686)

Referencias• S. Tremaine, “Is the Solar System Stable?”, IAS Newsletter, verano 2011,

pp. 1, 6

• V. I. Arnold, Huygens and Barrow, Newton and Hooke, (Birkhäuser, 1990)

• M. C. Gutzwiller, “Moon-Earth-Sun: The oldest three-body problem”, Rev. Mod. Phys. 70, 589 (1998)

• P. Holmes, “Poincaré, Celestial Mechanics, Dynamical Systems Theory, and ‘Chaos’ ”, Phys. Rep. 193, 137 (1990)

• M. Nauenberg, “Newton's perturbation methods for the three-body problem and their application to lunar motion'', en Isaac Newton's Natural Philosophy, eds. J. Z. Buchwald e I. B. Cohen, (MIT Press, 2001), pp. 189-224

• AJ, “The Mechanical Career of Councillor Orffyreus, Confidence Man”, Am. J. Phys. 81, 421 (2013) [arXiv:1301.3097 [physics.hist-ph]]